CircleBundleTransformation

1. A transformation related to circle bundles

Given two points {A(r,0),B(-r,0)}, taken on the x-axis symmetrically to the origin and a direction e(cos(u),sin(u)) define a transformation as follows:
- For each point X not lying on the x-axis consider the circle member c_X(x,y) of the bundle (of all circles through the two points) generated by the circle (x²+y²)-r²=0 and the line y=0.
- Then construct Y to be the other intersection point of c_X with the line {X+te} through X parallel to e.
[1] The transformation Y=F(X) is well defined for every point of the plane except the x-axis.
[2] It is involutive satifying obviously F² = 1.
[3] The fixed points of F are on a rectangular hyperbola o_X passing through {A,B}.
[4] The hyperbola o_X belongs to the family of conics (which are all rectangular hyperbolas) generated by the conic c₁(x,y)=x²-y²-r²=0 and the degenerate conic c₂(x,y) = xy = 0 (product of axes).
[5] o_X passes through point C = r(Je), where J denotes the positive rotation by a right angle.

[6] F restricted on each member c_X of the circle bundle leaves it invariant and coincides there with the reflexion along the line L_X passing through the center of c_X and being orthogonal to direction (e).

Remark Last property suggests a way to generate rectangular hyperbolas using circle bundles:
Fix a direction (Je) and for each member c_X of the circle bundle draw on either sides of the center I_X segments parallel to the selected direction at distance r_X from the center (r_X being the radius of c_X). The end-points of these segments describe a rectangular hyperbola. The hyperbola has center the origin O, passes through {A,B} and its axes are parallel to the bisectors of the angle between the x-axis and the Je-direction (see PowerGeneral.html ).

The proofs follow from some easy calculations.

This is the transformation Y=F(X). The fixed points result by equating the numerator to zero:
e₂(x²-y²-r²) - 2e₁(xy) = 0.
All statements follow from the form of this expression and some further easy calculations.

2. Lines mapping to hyperbolas

The transformation Y=F(X), defined above, maps non-horizontal lines of the plane to hyperbolas.

This can be seen by using the above calculations. Without loss of generality, for non-horizontal lines I assume that they are described by some parametric equation of the form
a + tb,
where a is a point of the x-axis a=(s,0) and b=(b₁,b₂) is a unit vector.
Assuming e to be a unit vector and taking inner products, last equation of the previous paragraph implies:

Last equation gives the desired result. In fact if Y=(x,y) then t=0 defines the line:
f(x,y)=(Y,Je)-(a,Je)=0.
The bracket represents another line g(x,y)=0 and on the right is a constant c. Thus last equation can be written in the form:
f(x,y)*g(x,y)=c,
where f(x,y)=0, g(x,y)=0 are the equations of two lines. Transforming to a coordinate system with these lines as axes we recognise that the equation transforms to x'y'=c', representing an hyperbola (see HyperbolaWRAsymptotics.html ).

The discussion continues in the file CircleBundleTransformationHyperbola.html , where I examine some geometric characteristics of these hyperbolas.

3. Lines mapping to parabolas

The transformation Y=F(X), defined above, maps horizontal lines of the plane to parabolas.

This time horizontal lines are described in the form
a + tb,
where a is a point of the y-axis a=(0,a₂) and b=(1,0).
Assuming e to be a unit vector and taking inner products, I use again last equation of the first paragraph.
This time the line is described through X=(a₂,t) and t simplyfies to t = -[(Y,Je)-a₂e₁]/e₂. Last equation of the first paragraph becomes
a₂(Y,e) = t²e₂ - ta₂e₁ - e₂r².
This, for Y=(x,y) and changing the axes of coordinates to y'=(Y,e) and x'=(Y,Je) transforms to an equation of the form
y' = ux'² + vx' + w, for appropriate coefficients {u,v,w}, representing a parabola as claimed.
Remark From the form of the equation follows that the axis of this parabola is in the (e) direction.

Some further discussion on the geometric characteristics of these parabolas is contained in the file CircleBundleTransformationParabola.html .

4. The fixed-point hyperbola and involutions

The transformation Y=F(X), defined above, maps lines L parallel to (e) to themselves and restricted there defines a homographic involution on L which coincides with the Desargues conjugation induced by the circle bundle on the line (see DesarguesInvolution.html ).
The fixed points of this involution are the intersection points of Line L with the rectangular hyperbola of the first paragraph.
This is an immediate consequence of the definition of F and the facts on involutions on lines generated by their intersections with circle-bundles discussed in Involution.html .

1. A transformation related to circle bundles

2. Lines mapping to hyperbolas

3. Lines mapping to parabolas

4. The fixed-point hyperbola and involutions

See Also